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Calculus 3 (Math 221) Notes and Study Guide

Brenton LeMesurier

Section 6.1 Vector Fields

Updated 2025-04-10; just correcting a typo in Definition 6.1.2.

References.

Topics.

Subsection 6.1.1 Definitions

Many physical quantities are described by vectors (like velocity, force) and depend on position in space or on a surface. The mathematical description of these is a vector field:

Definition 6.1.1. 2D Vector Field.

For \(D\) a set in \(\mathbb{R}^2\) (a plane region), a vector field on \(\mathbb{R}^2\) is a function \(\vec{F}\) that assigns to each point \((x,y)\) in \(D\) a two-dimensional vector \(\vec{F}(x,y)\text{.}\)
Such a vector field can be described with component functions as \(\vec{F} = P\veci + Q\vecj\text{;}\) that is
\begin{equation*} \vec{F}(x,y) = P(x,y)\veci + Q(x,y)\vecj = \vector{P(x,y),Q(x,y)} \end{equation*}
Scalar valued functions of several variables like \(P\) and \(Q\) are sometimes called scalar fields to distinguish them from vector fields.
There is an unsurprising 3D version:

Definition 6.1.2. 3D Vector Field.

For \(E\) a set in \(\mathbb{R}^3\text{,}\) a vector field on \(\mathbb{R}^3\) is a function \(\vec{F}\) that assigns to each point \((x,y,z)\) in \(E\) a three-dimensional vector \(\vec{F}(x,y,z)\text{.}\)
Again such a vector field can be described with component functions, so
\begin{equation*} \vec{F}(x,y,z) = P(x,y,z)\veci + Q(x,y,z)\vecj + R(x,y,z)\veck. \end{equation*}

Subsection 6.1.2 Gradient Vector Fields, or Conservative Vector Fields

Vector fields often arise as the gradient of a scalar function: for a function \(f\) of two variables the gradient
\begin{equation} \del f(x,y) = \vector{\partial_x f, \partial_y f} = \vector{f_x,f_y}\tag{6.1.1} \end{equation}
introduced in Subsection 4.6.2, The Gradient Vector is a vector field on \(\mathbb{R}^2\text{,}\) and likewise for \(f\) a function of three variables, its gradient is a vector field on \(\mathbb{R}^3\text{:}\)
\begin{equation} \del f(x,y,z) = \vector{f_x,f_y,f_z}\tag{6.1.2} \end{equation}

Remark 6.1.3.

Here we use the alternative notation “\(\partial_x\)” etc. for partial derivatives, which was introduced in (4.3.3) and (4.3.4).
In addition to the notations
\begin{equation} \Grad f = \nabla f = \del f\tag{6.1.3} \end{equation}
seen in Subsection 4.6.2, the symbol \(\nabla\) or \(\del\) is sometimes treated as the fake vector
\begin{equation} \del = \vector{\partial_x, \partial_y}\tag{6.1.4} \end{equation}
in the plane, and
\begin{equation} \del = \vector{\partial_x, \partial_y, \partial_z}\tag{6.1.5} \end{equation}
in space, which is “consistent” if you treat juxtaposition a bit like multiplication:
\begin{equation*} \del f = \vector{\partial_x, \partial_y} f = \vector{\partial_x f, \partial_y f} \end{equation*}
and so on.
This “vector” notation will be useful later, from Section 6.5, Divergence and Curl onward.

Definition 6.1.4.

A vector field \(\vec{F}\) is called a gradient vector field or conservative vector field if it is the gradient of some scalar function; that is, if there is a function \(f\) such that \(\del f = \vec{F}\text{.}\)
Function \(f\) is then called a potential function for \(\vec{F}\text{.}\)

Subsection 6.1.3 The Cross-Partial Property and Non-Conservative Vector Fields

For a conservative 2D vector field \(\vec{F} = \vector{P, Q}, = \vector{f_x, f_y}\text{,}\) look at the cross partial derivatives:
\begin{equation*} P_y = (f_x)_y = (f_y)_x = Q_x, \text{ so } Q_x - P_y = \partial_x Q - \partial_y P = 0 \end{equation*}
(recalling Clairaut’s Theorem 4.3.2 from [cross-reference to target(s) "section-partial-derivatives" missing or not unique])
This can be extended to 3D vector fields, and can be written in a suggestive mock-determinant notation that we will see more of in Section 6.5:

Exercise 6.1.6. A non-conservative vector field.

Verify that \(\vec{F}(x,y) = \vector{-y, x}\) is not conservative.

Exercise 6.1.7. A non-conservative vector field that has the cross-partial property.

Verify that the vector field
\begin{equation*} \ds\vec{F}(x,y) = \vector{-\frac{y}{x^2+y^2}, \frac{x}{x^2+y^2}} \end{equation*}
has the cross-partial property (6.1.6).
However, it will be seen in Section 6.3 that is not a gradient vector field on its natural domain of all points in \(\reals^2\) except for the hole at the origin.

Study Guide.

Study Section 6.1 of Calculus Volume 3
 6 
openstax.org/books/calculus-volume-3/pages/6-1-vector-fields
; in particular
  • The Definition and Theorems.
  • Examples 1–5 and 10–13, and the Checkpoints following each.
  • The following exercises (for ranges, do at least one in each range): 1, 2, 15–20, 22–24.
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